Let us try to use the original definition of differentiability: we would like to call the function f(X) = f(x, y, z) differentiable in a point C = (a, b, c) if the limit of the difference quotient
- ( f(X) - f(C) ) / (X - C)
as X approaches C exists. The numerator is well-defined, since f(X) and f(C) are real numbers, so that their difference can be computed as usual. For the denominator we can define the difference of X - C = (x, y, z) - (a, b, c) by taking the differences in each component, i.e. X - C = ( x - a, y - b, z - c). However, we have problems defining division. To make sense of the above difference quotient we must know how to define the quotient of a real number and a vector X = (x, y, z).
There is no satisfying definition of such a quotient. Therefore, we can not use this difference quotient in this situation to define 'derivative'.
To avoid this problem, we need a single real number in the denominator as well. Hence, we could look, for example, at the quotient
- ( f(X) - f(C) / (x - a)
where X = (x, y, z) and C = (a, b, c) and take the limit as x approaches c with all other variables fixed. This concept, somewhat modified, is known as partial derivatives.
Thus, we can use our original definition of differentiability to define partial derivatives, but not to define 'the derivative' of a function of more than one variable.
Note: There is one notable exception - the space of complex numbers. Recall that a complex number z = x + i y consists of a real and an imaginary part, where i is the square root of -1. Thus, a complex number z = x + i y can be identified with the tuple z = (x, y). And there is a way in two dimensional space to define division (and multiplication) in a meaningful way (how ?). Therefore, if we identify two-dimensional real space with the space of complex numbers, we can use the original definition of derivative to define the complex derivative of a complex function.